Statistical Preliminaries Stony Brook University CSE545, Fall 2016
Random Variables X : A mapping from Ω to ℝ that describes the question we care about in practice. 2
Random Variables X : A mapping from Ω to ℝ that describes the question we care about in practice. Example: Ω = 5 coin tosses = {<HHHHH>, <HHHHT>, <HHHTH>, <HHHTH>…} We may just care about how many tails? Thus, X(<HHHHH>) = 0 X(<HHHTH>) = 1 X(<TTTHT>) = 4 X(<HTTTT>) = 4 X only has 6 possible values: 0, 1, 2, 3, 4, 5 What is the probability that we end up with k = 4 tails? P (X = k ) := P ( {ω : X(ω) = k} ) where ω ∊ Ω 3
Random Variables X : A mapping from Ω to ℝ that describes the question we care about in practice. Example: Ω = 5 coin tosses = {<HHHHH>, <HHHHT>, <HHHTH>, <HHHTH>…} We may just care about how many tails? Thus, X(<HHHHH>) = 0 X(<HHHTH>) = 1 X(<TTTHT>) = 4 X(<HTTTT>) = 4 X only has 6 possible values: 0, 1, 2, 3, 4, 5 What is the probability that we end up with k = 4 tails? P (X = k ) := P ( {ω : X(ω) = k} ) where ω ∊ Ω X(ω) = 4 for 5 out of 32 sets in Ω . Thus, assuming a fair coin, P (X = 4) = 5/32 (Not a variable, but a function that we end up notating a lot like a variable) 4
Random Variables X : A mapping from Ω to ℝ that describes the question we care about in practice. Example: Ω = 5 coin tosses = { <HHHHH>, <HHHHT>, <HHHTH>, <HHHTH> …} We may just care about how many tails? Thus, X(<HHHHH>) = 0 X is a discrete random variable X(<HHHTH>) = 1 if it takes only a countable X(<TTTHT>) = 4 number of values. X(<HTTTT>) = 4 X only has 6 possible values: 0, 1, 2, 3, 4, 5 What is the probability that we end up with k = 4 tails? P (X = k ) := P ( {ω : X(ω) = k} ) where ω ∊ Ω X(ω) = 4 for 5 out of 32 sets in Ω . Thus, assuming a fair coin, P (X = 4) = 5/32 (Not a variable, but a function that we end up notating a lot like a variable) 5
Random Variables X : A mapping from Ω to ℝ that describes the question we care about in practice. X is a continuous random variable if it X is a discrete random variable can take on an infinite number of if it takes only a countable values between any two given values. number of values. 6
Random Variables X : A mapping from Ω to ℝ that describes the question we care about in practice. Example: Ω = inches of snowfall = [0, ∞) ⊆ ℝ X amount of inches in a snowstorm X is a continuous random variable if it can take on an infinite number of X (ω) = ω values between any two given values. What is the probability we receive (at least) a inches? P (X ≥ a) := P ( {ω : X(ω) ≥ a} ) What is the probability we receive between a and b inches? P (a ≤ X ≤ b) := P ( {ω : a ≤ X(ω) ≤ b} ) 7
Random Variables X : A mapping from Ω to ℝ that describes the question we care about in practice. Example: Ω = inches of snowfall = [0, ∞) ⊆ ℝ X amount of inches in a snowstorm X is a continuous random variable if it can take on an infinite number of X (ω) = ω P (X = i) := 0, for all i ∊ Ω values between any two given values. (probability of receiving exactly i What is the probability we receive (at least) a inches? inches of snowfall is zero) P (X ≥ a) := P ( {ω : X(ω) ≥ a} ) What is the probability we receive between a and b inches? P (a ≤ X ≤ b) := P ( {ω : a ≤ X(ω) ≤ b} ) 8
Random Variables, Revisited X : A mapping from Ω to ℝ that describes the question we care about in practice. Example: Ω = inches of snowfall = [0, ∞) ⊆ ℝ X amount of inches in a snowstorm X is a continuous random variable if it can take on an infinite number of X (ω) = ω P (X = i) := 0, for all i ∊ Ω values between any two given values. (probability of receiving exactly i What is the probability we receive (at least) a inches? inches of snowfall is zero) P (X ≥ a) := P ( {ω : X(ω) ≥ a} ) How to model? What is the probability we receive between a and b inches? P (a ≤ X ≤ b) := P ( {ω : a ≤ X(ω) ≥ b} ) 9
Continuous Random Variables Discretize them! (group into discrete bins) How to model? 10
Continuous Random Variables P (bin=8) = .32 P (bin=12) = .08 But aren’t we throwing away information? 11
Continuous Random Variables 12
Continuous Random Variables X is a continuous random variable if it can take on an infinite number of values between any two given values. X is a continuous random variable if there exists a function fx such that: 13
Continuous Random Variables X is a continuous random variable if it can take on an infinite number of values between any two given values. X is a continuous random variable if there exists a function fx such that: fx : “probability density function” (pdf) 14
Continuous Random Variables 15
Continuous Random Variables 16
Continuous Random Variables Common Trap ● does not yield a probability ○ does ○ � may be anything ( ℝ ) ■ thus, may be > 1 17
Continuous Random Variables A Common Probability Density Function 18
Continuous Random Variables Common pdf s: Normal( μ , σ 2 ) = 19
Continuous Random Variables Common pdf s: Normal( μ , σ 2 ) = μ : mean (or “center”) = expectation σ 2 : variance, σ : standard deviation 20
Continuous Random Variables Common pdf s: Normal( μ , σ 2 ) Credit: Wikipedia = μ : mean (or “center”) = expectation σ 2 : variance, σ : standard deviation 21
Continuous Random Variables Common pdf s: Normal( μ , σ 2 ) X ~ Normal( μ , σ 2 ), examples: ● height ● intelligence/ability ● measurement error ● averages (or sum) of lots of random variables 22
Continuous Random Variables Common pdf s: Normal(0, 1) (“standard normal”) How to “standardize” any normal distribution: ● subtract the mean, μ (aka “mean centering”) ● divide by the standard deviation, σ z = (x - μ ) / σ , (aka “z score”) Credit: MIT Open Courseware: Probability and Statistics 23
Continuous Random Variables Common pdf s: Normal(0, 1) Credit: MIT Open Courseware: Probability and Statistics 24
Cumulative Distribution Function For a given random variable X, the cumulative distribution function (CDF), Uniform Fx: ℝ → [0, 1] , is defined by: Normal 25
Cumulative Distribution Function For a given random variable X, the cumulative distribution function (CDF), Uniform Fx: ℝ → [0, 1] , is defined by: Pro: yields a probability! Exponential Con: Not intuitively interpretable. Normal 26
Random Variables, Revisited X : A mapping from Ω to ℝ that describes the question we care about in practice. X is a continuous random variable if it X is a discrete random variable can take on an infinite number of if it takes only a countable values between any two given values. number of values. 27
Discrete Random Variables For a given random variable X, the cumulative distribution function (CDF), Fx: ℝ → [0, 1] , is defined by: X is a discrete random variable if it takes only a countable number of values. 28
Discrete Random Variables For a given random variable X, the cumulative distribution function (CDF), Fx: ℝ → [0, 1] , is defined by: X is a discrete random variable if it takes only a countable number of values. Binomial (n, p) (like normal) 29
Discrete Random Variables Binomial (n, p) For a given random variable X, the cumulative distribution function (CDF), Fx: ℝ → [0, 1] , is defined by: X is a discrete random variable if it takes only a countable number of values. For a given discrete random variable X, probability mass function ( pmf ), fx: ℝ → [0, 1] , is defined by: 30
Discrete Random Variables Binomial (n, p) Two Common Discrete Random Variables ● Binomial(n, p) example: number of heads after n coin flips (p, probability of heads) ● Bernoulli(p) = Binomial(1, p) example: one trial of success or failure 31
Hypothesis Testing Hypothesis -- something one asserts to be true. Classical Approach: H 0 : null hypothesis -- some “default” value; “null” => nothing changes H 1 : the alternative -- the opposite of the null => a change or a difference
Hypothesis Testing Hypothesis -- something one asserts to be true. Classical Approach: H 0 : null hypothesis -- some “default” value; “null” => nothing changes H 1 : the alternative -- the opposite of the null => a change or a difference Goal: Use probability to determine if we can “reject the null”( H 0 ) in favor of H 1 . “ There is less than a 5% chance that the null is true” (i.e. 95% alternative is true). Example: Hypothesize a coin is biased. H 0 : the coin is not biased (i.e. flipping n times results in a Binomial(n, 0.5))
Hypothesis Testing Hypothesis -- something one asserts to be true. H 0 : null hypothesis -- some “default” value (usually that one’s hypothesis is false) Classical Approach: H 1 : the alternative -- usually that one’s “hypothesis” is true H 0 : null hypothesis -- some “default” value (usually that one’s hypothesis is false) More formally: Let X be a random variable and let R be the range of X. R reject ⊂ R is the rejection region. If X ∊ R reject then we reject the null.
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